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Mirrors > Home > ILE Home > Th. List > tposfn2 | GIF version |
Description: The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposfn2 | ⊢ (Rel A → (𝐹 Fn A → tpos 𝐹 Fn ◡A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposfun 5816 | . . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹) | |
2 | 1 | a1i 9 | . . 3 ⊢ (Rel A → (Fun 𝐹 → Fun tpos 𝐹)) |
3 | dmtpos 5812 | . . . . . 6 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (dom 𝐹 = A → (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)) |
5 | releq 4365 | . . . . 5 ⊢ (dom 𝐹 = A → (Rel dom 𝐹 ↔ Rel A)) | |
6 | cnveq 4452 | . . . . . 6 ⊢ (dom 𝐹 = A → ◡dom 𝐹 = ◡A) | |
7 | 6 | eqeq2d 2048 | . . . . 5 ⊢ (dom 𝐹 = A → (dom tpos 𝐹 = ◡dom 𝐹 ↔ dom tpos 𝐹 = ◡A)) |
8 | 4, 5, 7 | 3imtr3d 191 | . . . 4 ⊢ (dom 𝐹 = A → (Rel A → dom tpos 𝐹 = ◡A)) |
9 | 8 | com12 27 | . . 3 ⊢ (Rel A → (dom 𝐹 = A → dom tpos 𝐹 = ◡A)) |
10 | 2, 9 | anim12d 318 | . 2 ⊢ (Rel A → ((Fun 𝐹 ∧ dom 𝐹 = A) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡A))) |
11 | df-fn 4848 | . 2 ⊢ (𝐹 Fn A ↔ (Fun 𝐹 ∧ dom 𝐹 = A)) | |
12 | df-fn 4848 | . 2 ⊢ (tpos 𝐹 Fn ◡A ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡A)) | |
13 | 10, 11, 12 | 3imtr4g 194 | 1 ⊢ (Rel A → (𝐹 Fn A → tpos 𝐹 Fn ◡A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ◡ccnv 4287 dom cdm 4288 Rel wrel 4293 Fun wfun 4839 Fn wfn 4840 tpos ctpos 5800 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 df-tpos 5801 |
This theorem is referenced by: tposfo2 5823 tpos0 5830 |
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